There is a mode called direct law that effectively gives the pilots full authority. This isn’t fly-by-wire in the sense that fighters are, where they’re designed to be unstable and are essentially unflyable without computer assistance. The fly-by-wire in airliners is designed to provide protection from flight limits such as g loading, stalling etc, and to make flying easier under normal conditions. An airbus will auto-trim for you, so you use the sidestick to achieve the attitude you want, let go and the attitude will stay there. If you remove the fly-by-wire aspect it just turns into a normal aeroplane without protections, it doesn’t become in any way unflyable.
The problem with all simulators, from full motion Cat D sims used for airline training to PC based flight sims, is that they don’t have the data to be able to simulate conditions at the edges of the flight envelope. There is currently no way to accurately simulate what happened to AF447 other than to take a couple of test pilots up in a real aeroplane. That may seem criminal. Surely you should be able to accurately train pilots in recovery from a full stall. Well it would be nice if you could but the technology is not available and so instead they give you full stall training in small aircraft but go on to say DO NOT STALL! To help pilots with this they design the large aeroplanes to not let them stall. They put stick shakers in to alert of an impending stall and allow recovery before the stall occurs, and they put stick pushers in that physically push the control column forward which does two things, it prevents a full stall from occurring, and it provides the pilot with a simulation of the stalling characteristics of docile training aircraft. In Airbus’s case they have a computer that moderates the pilots inputs and if it sees anything it doesn’t like it says “nope, can’t do that sorry.” Having said all that, the pilots can ignore the stick shaker, they can fight the stick pusher (ala Colgan Air), and they can put an Airbus into Direct Law which turns it into a normal aeroplane with no protections. Also, if the aeroplane is faulty and multiple redundant systems fail, all bets are off. There are no good answers when all the backups have failed.
You would normally power out of a stall warning (i.e. when the stick shaker goes but the aeroplane hasn’t stalled yet), but thrust gives a nose up pitching moment and if you are having trouble getting the nose down to reduce the angle of attack one thing you’d want to do is get rid of the thrust.
At the start of the problems the aircraft went into Alternate Law and the auto thrust disconnected, Alt Law doesn’t provide any protections other than to pitch the nose up gently at high Mach numbers and to pitch the nose down gently at low airspeeds, both of those protections can be over-ridden by the pilots. In short, they should not have had any issue fighting the aeroplane.
The engines aren’t positioned in a way to put airflow over the control surfaces. Also the engines are most efficient at high RPMs and the airframe is most efficient at a particular moderate indicated airspeed. Indicated airspeed drops rapidly compared to true airspeed as altitude increases, for example an indicated speed of 250 knots is approximately a true airspeed of 380 knots at 30’000’. The general concept with cruising is to fly high where the thrust produced from a high engine RPM coincides with an efficient airspeed. There are margins though, you have margins above the stall and margins below speed of sound related effects. The margins may not be as large as what you might expect but they are by no means small. At all times the aircraft should be able to manoeuvre and stay a safe margin above the stall and below the mach buffet speeds. If turbulence is forecast then you fly lower where you have even more margin. When the pilots said, before the accident, that they couldn’t get to a higher level, it doesn’t necessarily mean they literally didn’t have the performance to get there, it may mean they can’t do it and maintain their safety margins. As the aircraft gets lighter the stalling speed drops and they can go higher, they also have more excess power available.
I think something that might be confusing to the layman is that nothing in flying is simple. The airspeed indicator doesn’t actually tell you your airspeed, the altimeter doesn’t tell you how high you are, the compass doesn’t tell you which way north is, power doesn’t just make you go faster, it also pitches the nose up, rolling doesn’t just roll, it also yaws in the opposite direction, and yawing rolls in the opposite direction, changing power can produce yaw as well, the stall speed is only valid under a specific set of conditions, and there is not normally any direct indication to the pilots of how close to the stalling angle they are. basically none of the indications you have are direct, and everything you do has a secondary effect. That’s not to say it’s difficult, because it’s not, but it is not very intuitive.